Energy

The work in this chapter is the most technically demanding in this book, and also the area most needing follow-up work … by people like you! As I note in Manifesto , the fact that economics has persisted for almost a century (Cobb and Douglas 1928; Leontief 1944, 1946b; Leontief 1946a; Leontief 1936) with models of production in which energy plays no role is, arguably, the Original Sin of Economics that has resulted in it being the misleading miasma that it is today. But escaping from that miasma is difficult, as I found as I worked with Matheus Grasselli and Tim Garrett to derive the models outlined here.

The starting point, though, was simple enough. Both Neoclassical and Post Keynesian mathematical models of production functions treated output as a function of inputs of Labor and Capital:

However, nothing can be produced without energy and matter inputs as well. The input-output approach to modelling production, pioneered by Leontief (Leontief 1936), did explicitly include inputs of both energy and raw materials (as well as other commodities) to produce output, but in practice, this method was generally implemented in an equilibrium framework in “Computable General Equilibrium” (CGE) models, when the equilibrium of an input-output matrix is unstable.[77] After the “Rational Expectations Revolution”, Neoclassicals largely abandoned CGE modeling in favour of single commodity modeling, based on the Ramsey growth model (Ramsey 1928). The Cobb-Douglas Production Function (CDPF) ruled supreme in these models, and portrayed output as being produced by combining technology , labour and capital : 1   A L  K

Post-Keynesian aggregate production form of the Leontief input-output model, which Goodwin used in his cyclical growth model (Goodwin 1967), and I used in my model of Minsky’s Financial Instability Hypothesis, is:

  v  Here is normally described as the “Capital to Output ratio” (see Figure 69 on page 48), while is called “Labor Productivity”—though I challenge both these labels later. v T

Neither aggregate production function explicitly include either matter or energy, something which mainstream economists largely ignored until the publication of the Limits to Growth (Meadows, Randers, and Meadows 1972). Then, faced with a rival technology—system dynamics—they tried to develop a Neoclassical riposte. Stiglitz (Stiglitz 1974a; Stiglitz 1974b) and Solow (Solow 1974a) both proposed modified Cobb-Douglas functions of the form (Solow 1974a, p. 35, Equation 6):

Here R stood for “Resources”, which include energy.[78] It is treated as a third input on equal footing with Labor and Capital.

This didn’t make sense to me, for two reasons. Firstly, it implied that energy could be added to a production process independently of labor and capital—say, by hitting a factory with a bolt of lightning—and thus producing output. But this was more likely to turn the factory into a smouldering ruin. Secondly, it implied that Labor and Capital could both function without energy—which of course they can’t. Figure 206 both portrays and satirizes this approach.

Even far superior attempts to engage with the role of energy in production, like the work of Kümmel and Ayres (Kümmel, Ayres, and Lindenberger 2010; Kummel 2011; Lindenberger and Kümmel 2011; Kümmel, Lindenberger, and Weiser 2015), used a similar formulation where Capital, Labor and Energy were put on an equal footing. One step in the derivation of their LinEx production function was the introduction of a dimensionless specification of a production function, which again put Labor, Capital and Energy on an equal footing. Equation (89) shows equations 39 and 50 from (Kümmel, Ayres, and Lindenberger 2010, pp. 162,166)

What was needed was a formulation which made energy absolutely essential to the production process, and didn’t pretend that it could be added independently of both labour and machinery.

Figure 209: Treating energy as an equivalent independent input to labour and capital

I was cogitating over this dilemma one evening while walking through Bob Ayres’s apartment in Paris—which was full of statues—when the quip “Capital without energy is a sculpture; labor without energy is a corpse” flashed into my mind. This insight revealed that the correct form for incorporating energy in production wasn’t Equation (89) and Figure 206, but equation (90) and Figure 207. Energy is an input to both machinery and labour, without which they can’t do useful work:

In doing useful work, waste is also necessarily generated—an application, in a very limited sense, of the Second Law of Thermodynamics. So the inputs to Labour and Capital are (different forms of) energy, and the outputs are materials transformed from non-usable inputs to usable commodities, plus waste.

Figure 210: Labour and capital both need energy inputs to produce output (which inevitably produces waste)

The easiest way to develop a mathematical model of production out of this insight was to treat both and as being equal to the product of the units of each ( and ), times the annual energy consumption of each ( and ), times how efficiently those inputs were turned into useful work ( K(E) and L(E) ): K L K L E E AK AL K E  K E  e

I fed this into the Capital and Labour components of the Cobb-Douglas Production Function (minus what soon transpired to be the superfluous ): α 1 −α = A ⋅

Rearranging this led to the expression in Equation (93), where the last two components are the standard expressions for capital and labour in the CDPF:

The first component—the energy consumption of the typical worker, times how much of that energy is turned into useful work in production—can be treated as a constant: the capacity for a worker to put energy into useful work hasn’t varied since humans evolved, and is roughly 100 Watts. The second is the energy input to the “representative machine” at a given time, multiplied by how much of that energy is turned into useful work. The energy consumption of the “representative machine”

has risen from the tonnes of fuel per day that powered James Watt’s steam engine to the tonnes per second that fuel Elon Musk’s rockets. The efficiency with which machines turn energy into useful work is an unknown scalar bounded by (0,1) . Treat the product )1−α as a constant and reserve the exponents for factors that actually change over time: . Then our energyL L Kα L (E ∙A ∙A C modified Cobb-Douglas Production Function is equation (94): K E ,K, L

t = ⋅ ⋅ ⋅ L K Derived this way, the “total factor productivity” term is actually a constant times the energy input to the “representative machine” of a given time. A

This is the form in which I published this work (with Bob Ayres and Russell Standish) in “A Note on the Role of Energy in Production” (Keen, Ayres, and Standish 2019, p. 44). But we only used the Cobb-Douglas Production Function in the probably forlorn hope that some Neoclassicals might therefore read the paper. The real basis for modelling the role of energy in production properly is the “Leontief Production Function” used by Post Keynesians (equation (95)):

On the other hand, the Cobb-Douglas Production Function belongs in the dustbin of the history of economic thought.

10.1 Forget the “Cobb-Douglas Production Function” (an optional read)

Neoclassicals take great solace in the fact that their preferred aggregate production function fits the national data so well:

I have always found the high R[2] reassuring when I teach the Solow growth model. Surely, a low R[2] in this regression would have shaken my faith. (Mankiw 1997, p. 104)

This is doubly so, because the model also encapsulates the Neoclassical belief that the real wage is the marginal productivity of labor, and the rate of profit is the marginal productivity of capital. The fact that the empirically measured Cobb-Douglas exponents are very close to the national income shares of labour and capital played a major role in the acceptance of the Cobb Douglas by Neoclassical economists:

aggregate production functions apparently work nevertheless and do so in a way which is prima facie not easy to explain. It is easy enough to understand why, in economies in which things move more or less together, a relationship giving an aggregate measure of output as dependent on aggregate measures of capital and labor should give a good fit when applied to the data. What is not so easy to explain is the fact that the marginal product of labor in such an estimated relationship ap- pears to give a reasonably good explanation of wages as well. In its simplest form, this puzzle is set by a remark which Solow once made to me that, had Douglas found labor's share to be 25 per cent and capital's 75 per cent instead of the other way around, we would not now be discussing aggregate production functions . (Fisher 1971, p. 305)

Tragically, in one of the most insightful and witty papers in the history of economics,[79] “The Humbug Production Function”, Anwar Shaikh (Shaikh 1974) gave the explanation that Fisher craved—and it wasn’t one that Fisher would have enjoyed. The Cobb-Douglas Production Function is just a tautology . It simply restates, in exponential rather than additive form, the identity that “Income equals Wages plus Profits” under conditions of relatively constant income shares:

Shaikh demonstrated that the Cobb–Douglas is simply an (anti-)logarithmic transformation of the income identity under the assumption that relative income shares are constant. (Carter 2011, p. 259)

Therefore, regressing the Cobb-Douglas Production Function against national income data is like regressing Y against Y: of course you’ll get a high correlation. That correlation falls below 100% only to the extent to which its assumptions—such as a uniform wage rate and constancy of income shares—deviate from actual conditions.

Bring in income shares—the proportion of income going to workers and capitalists respectively—by multiplying each fraction by the “missing ingredient”: multiply the first term by ⁄ , the second by ⁄ and so on: L 1 d L w d w L d K r d r K d w     

Group the terms so that income shares multiply each differential:

“Percentage” rates of change can be expressed as the differential of the logs, so that

And likewise for the other differentials in (101):

       dt dt dt dt dt At this point, we assume that income shares and are constant. They do change over time— that was the basis of the Goodwin model[80] —but relatively slowly compared to employment, wages, α 1 −α capital and the rate of return on capital, as codified in Kaldor’s stylized facts:

the share of wages and the share of profits in the national income has shown a remarkable constancy in " developed " capitalist economies of the United States and the United Kingdom since the second half of the nineteenth century. (Kaldor 1957, pp. 591-92)

1957, pp. 591-92) Neoclassical modelers also treat as a constant in their models. So we can do the same, and then integrate both sides, with integration being the inverse of differentiation: α

A constant multiplying the logarithm of a variable is the same as the logarithm of the variable raised to the power of that constant: ) and so on, so that 1  1    ln Y α∙LS(w) = LS(w  ln w lnα L ln K ln r (105)

Take exponentials of both sides:

This is almost the “Cobb-Douglas Production Function”: the only difference is that Cobb and Douglas began with a constant in the place of , while later Neoclassicals use a time-varying , which they call “total factor productivity”—and which, as explained previously, is actually the energy w1−α ∙gα A(A) consumption level of the “representative machine”:

It’s no wonder, therefore, that the “Cobb Douglas Production Function” fits the empirical data on output and income distribution, since it can be derived from that data, under the not entirely false assumption that income shares are relatively constant.

Neoclassicals estimate as a residual from the time series for Labour and Capital—since the vast majority of them are not aware of Shaikh’s proof, and in typical Neoclassical fashion, those that are A think that Solow’s rejoinder to Shaikh (Solow 1974b) settled the dispute in their favour. But it didn’t (Shaikh 1980, 2005; Labini 1995).

There is a further weakness, pointed out by Mankiw and noted in Manifesto , that while the CDPF fits national data well with its exponent conforming to national income distribution data, this value for also results in predictions of relative economic performance that are disastrously bad: α

Because poor countries have about one-tenth the income of rich countries, they should have returns to capital that are about one hundred times as large. In particular, since the profit rate is about 10 percent per year in rich countries, it should be about 1,000 percent per year in poor countries. (Mankiw, Phelps, and Romer 1995, p. 287)

Another good reason to reject the CDPF is its assumed easy substitution of one input for another. This in itself is a dubious assumption—you can’t easily vary the labour and capital inputs into a production process—but in the context of energy, it is simply false. Energy can be used more or less efficiently, but there is no substituting for it. If you don’t have energy, you don’t have output, period. On this basis, the fixed coefficient formulation of the Leontief is more sensible. And, as the next section shows, it is easy to interpret the capital output ratio in the Leontief function as the efficiency with which energy is turned into useful work. The Leontief function has therefore implicitly contained the role of energy all along.

10.2 Generalizing the Leontief Production Function

Superficially, the Leontief Production Function has the same weakness as the Cobb-Douglas when it comes to the role of energy of energy in production—there isn’t one. Stating the Leontief in terms of a utilization of capital rate , a capital to output ratio , and an output to labour ratio , it is: K E Y ( t ) = u ⋅ v = a ⋅ L T (108) v In fact, it’s relatively easy to show that the capital to output ratio v, which has been treated simply as an empirical regularity with a fairly constant value of between 2 and 4 for most economies, is actually the inverse of : the efficiency with which machines turn energy into useful work. We have to start by defining what aggregate output AK actually is, mathematically, in a macroeconomic model. Economists treat it as just a number—a scalar—but the real question is “a number of what?”. It is not a pure number, but a dimensioned number: it is a number of identical Y “things”. These “things” are stylized universal commodities, which in the models can be consumed by workers as consumption items , or used as investment items , which are inputs to make machines, or “Capital” . The term for this (highly unrealistic) universal commodity is a “widget”. So C I in an aggregate macroeconomic model is the number of widgets produced per year. K In this same sense, we—my collaborators Matheus Grasselli (Grasselli and Costa Lima 2012; Grasselli Y and Maheshwari 2017; Grasselli and Nguyen-Huu 2018; Giraud and Grasselli 2019) and Tim Garrett (Garrett 2011, 2012a, 2012b, 2014, 2015) and I—introduced as the energy equivalent of : it was the amount of energy (measured in joules) contained in a widget, multiplied by the number of widgets produced per year . Q Y Y E Y Q = E ⋅ Y

We then equated Q to the energy converted into useful work by machinery, using equation (91):

We can now show the relationship between Q and Y, using equation (108):

K  eK  v If we now equate terms with the same dimensions—energy per year in the cases of and scalars in the cases of , we get, firstly, that is the inverse of : Y K E ,E K K 1 A , v A v

Secondly, the conversion factor between output in widgets and output in terms of energy (useful work) at any given time in this single-commodity world is the energy consumption level of the typical machine of that time:

The first finding was a surprise, but one that made intuitive sense once we realized it: the empirically-observed rough proportionality between output Y and capital stock K, which is an essential aspect of the Leontief model, actually represents the efficiency with which machines turn their energy inputs into useful work. In this sense, the Leontief model has always included a role for energy—it just wasn’t explicit. This then turns on its head the standard rendition of the capital to output ratio. This has been declining over time, somewhat inexplicably—see Figure 208:

_Figure 211: Capital output ratio from https://fred.stlouisfed.org/series/RKNANPUSA666NRUG#0_

However, from this energy-based perspective, what this actually shows is a rise over time in the efficiency with which machinery turns energy into useful work—or, also quite feasibly, an increase in the amount of GDP which is virtual or non-physical (neither commodities nor directly consumed energy, though of course virtual products—such as video games—require physical resources, including file servers and electricity). Though there is an increasing trend right from the start of the data, it becomes much stronger and more pronounced in the early 1980s, which coincides with the

development of the computer and the “virtual” economy it allows, the financialization of capitalism and the rise in what Marx would call “fictitious output” from “fictitious capital”,[81] and the start of US capital outsourcing production to China.

Figure 212: The efficiency with which energy is turned into useful work (GDP, or Y)

The rise in the ratio also supports to some degree the “decoupling” argument, that over time less and less of GDP is dependent on physical and energetic output—though it’s also important to put this in context: the dependence at the global level of output on energy remains extremely high (the data in Figure 209 comes solely from the USA). When one looks at the long-run global data (Figure 210), and especially data for the last half-century (Figure 211), the correlation between GDP and energy is extremely tight.

Figure 213: Global GDP and energy consumption since 1800__82

82 Data sources Pre-1960:

Energy https://themasites.pbl.nl/tridion/en/themasites/hyde/consumptiondata/totalenergy/index-2.html GDP https://www.rug.nl/ggdc/historicaldevelopment/maddison/releases/maddison-database-2010

Figure 214: Global GDP and Energy data since 1970__83

These are both rising trends which generates a spurious correlation of course, but the annual change data is also extremely tightly correlated, and almost 1 for 1—see Figure 212.

83 Data sources Post 1970: - - Energy https://data.oecd.org/energy/primary energy supply.htm GDP https://data.worldbank.org/indicator/NY.GDP.MKTP.KD

Figure 215: Annual change in GDP against change in energy (Correlation 0.83)

However, the rise in the GDP to energy ratio is also apparent at the global level since the 1970s—see Figure 213.

Figure 216: GDP in constant US$ divided by Energy in BTU

However, the long-term data shows that this is a reversal of the trend since the start of the industrial age—see Figure 214.

Figure 217: GDP to Energy since 1800

Interpretation of this long term trend in GDP to Energy is an open question—it quite possibly represents the change from non-fossil to fossil-fuel driven industry over the course of the 19th century. That said, the very tight fit between energy and GDP from the 20th century on, and especially for the period from 1970 till 2017, provides another strong argument for the Leontief Production Function as the proper tool to model the close to linear relationship between energy consumption and GDP.

10.3 A Goodwin model with Energy

One key element in the previous section was using dimensional analysis to unravel an equation— equation (111), where the first term on the left was dimensioned in units of energy per year, and the next term was a scalar. It therefore made sense to equate components in the equation with the same dimensions:

Dimensional analysis is an important technique in science and engineering to check the validity of a model, and it should be in economics too:

The consistent and correct use of dimensions is essential to scientific work involving mathematics. Their very existence creates the potential for errors: omitting them when they should be included, misusing them when they are included, and others. However, their existence also makes possible dimensional analysis, which can be a significant factor in avoiding error. In the equation , if y should have dimensions then so also should , and they should be identical to those of y. If y should not have them then neither should have E= f(. ) f(. ) them… An error revealed by a correctly performed dimensional analysis indicates f(. ) a fundamental problem. Therefore, the importance of dimensions for science can hardly be overstated. (Barnett II 2004, p. 95).

Economics has ignored dimensional analysis, as is obvious enough in the Cobb Douglas Production function itself. As Barnett points out, the dimensions of the function can only be made reasonable by ascribing a ludicrous set of dimensions to the term: A typical CD function is given by A(A) , in which: Q is the output variable; K and L are the capital and labor input variables, respectively; A, may be a constant or a variable; and, and Q = A∙K are the elasticity of output with respect toα ∙Lβ capital and with respect to labor, respectively… α β If dimensions are used correctly, output, capital, and labor each must have both magnitude and dimension(s), while and are pure numbers. Assume, for example, that: α β

• (1) Q is measured in widgets/elapsed time (wid/yr);
• (2) K is measured in units of machine-hours/elapsed time (caphr/yr); and,
• (3) L is measured in man-hours/elapsed time (manhr/yr). (Barnett II 2004, p. 96)

The only way to balance this equation in dimensional terms is for the A term to have crazy dimensions for something that Neoclassicals, not knowing of Shaikh’s critique, describe as “Total Factor Productivity”:

The “Cobb Douglas Production Function”, as well as being based on a tautology, is also dimensionally weird. What we need instead is a model of the biophysical processes by which inputs of energy, raw

materials and intermediate products are turned into usable physical products.[84] This chapter will take the first tentative steps towards this, in models in which energy plays a fundamental role. Our first pass was a model in which the inputs are energy, and the outputs are energy: the production process turns energy in a form that can’t be consumed by humans—say, coal—into one in which it can—say, electricity.

We started from the points established earlier about the Leontief Production Function (LPF), that by using the redefinition of K and L as means by which energy is used to perform useful work:

We can recast the standard LPF:

This is dimensionally consistent:

The Leontief Production Function in terms of energy per year is mapped across to the standard measure of GDP in Widgets per year by dividing by , where , the energy content of a widget: K K Y Q E E = E

E E Y K For simplicity, I’ll work with as in Goodwin’s original model.[85] We start from the definition of in terms of : E= 1 Q Q K  K E  e

Labour’s input also has to be converted into energy terms, where we treat the energy output of the representative worker as a constant:[86]

Labour is a derived demand in the Goodwin model: it is equal to the number of workers needed to operate the machines used to produce output. We therefore need to define a machine to worker ratio:

L L In the original Goodwin model, Goodwin used an output to labour ratio , which he assumed rose over time at a constant rate , and this was the same as the rate of growth of the capital to labour T ratio (since there was a linear relationship between output and capital). is therefore equivalent to α in (Goodwin 1967). As with Goodwin, we assume that this ratio rises exogenously over time, but as L well as giving it a less androgynous term ( rather than ), we use a less androgynous Greek letter k T kappa ( ) for its rate of growth: L k T  L

κ k = κ

The output to labour ratio in this model is more complicated, since it relates the useful energy from production to the energy input from labour. It therefore includes the dynamics of energy as well as of those of the capital to labour ratio:[87]

Once is defined, the rest of the model follows logically. The employment rate L is employment divided by population , which is assumed to grow at an exogenously given rate. Goodwin used for this rate; in keeping with our eponymous renaming of λ L N the capital to labour ratio, we use (the Greek equivalent of n) instead: β L ν

The employment rate determines the rate of change of wages:

The wage times Labour determines the wage bill, which determines profit:

Investment is profit minus depreciation:

Capital times the energy output of capital determines output Q in units of energy per year, closing the causal chain of the model:

10.3.1 Derivation

We start from the same system states as in the original Goodwin model, and then expand them out with the new definitions from equations (121) to (133). Firstly the derivation of : L λ̂

Therefore

The derivation of IG : K

Therefore:

This is strictly identical to the original Goodwin model form:

At this stage the inclusion of energy might look like “much ado about nothing”—see Figure 215.

Figure 218: Goodwin with energy in system state form

However, there are three ways in which this is an advance:

1. The previous empirical regularity of a reasonably constant capital to output ratio is now explained as the efficiency with which energy is converted into useful work;
2. The explicit use of energy in the derivation allows both waste production (consistent with the 2nd Law of Thermodynamics) and resource depletion to be added to the basic Goodwin model.

Point 3 above is covered by firstly defining waste energy as the complement to useful energy in Equation (133):

Secondly, to simulate resource depletion, we revise Equation (133) to include a factor based on the fraction of remaining fossil fuel reserves:

Depletion includes the use of energy in production, and the energy consumed by workers.

This extension is best shown in an absolute values model of Goodwin with energy. This model is shown in Equation (144) and simulated in Minsky in Figure 216 (the Minsky model includes a conversion of waste energy into waste matter, which can be degraded over time—we’re thinking of C 2here obviously).

Figure 219: A Minsky system dynamics model of energy in production and resource depletion

This explains the final figure in Manifesto , but it only scratches the surface of properly incorporating inputs from Nature into economic modelling. Though the previous model does introduce energy into the production function, its treatment of matter is too simplistic, with all the “heavy lifting” between matter and energy done by the conversion factor . A model of production entirely in terms of energy is also an extreme simplification. More realistically, energy is used in production to transform K E matter from less useful forms (raw materials) to more useful (finished products). The next section develops a model with both energy and matter inputs used to produce useful matter output. This model was derived in collaboration with my friends and research colleagues Tim Garrett, an atmospheric physicist, and Matheus Grasselli, a financial mathematician.

10.4 A Goodwin model with matter and energy

Our inspiration here was Hicks’s noble but unsuccessful attempt to build a dynamic model of a production economy in the paper "Wages and Interest: The Dynamic Problem" (Hicks 1935), where the output was bread—a consumer good. Though poorly known today, this paper was in fact the real origin of the IS-LM model, as Hicks admitted in 1981 in "IS-LM: An Explanation" (Hicks 1981).[88] The key evidence to which Hicks alluded was the section of the 1935 paper that used equilibrium in two markets to mean that equilibrium in a third could be assumed—and therefore the analysis could be simplified by omitting that market entirely: “An obvious result, so it would appear! But it conveys the less obvious message, that in order to determine the rate of interest, we need not examine that elusive thing, the "capital

In this paper, Hicks attempted to develop a dynamic theory of economics by reconciling the treatment of capital as a “factor of production” by J.B. Clark with its treatment as a produced means of production by Wicksell:

Most modern theories of capital fall into one or two classes. On the one hand, there is the "timeless" type of theory, which treats capital as a factor of production like any other. Such a theory is that of J. B. Clark. In practice, it assimilates capital to land, treating it as the inexhaustible provider of a regular stream of resources. On the other hand, there is the "period of production" theory of Bohm-Bawerk and Wicksell. This treats capital as "stored-up labour"— labour stored up in the past . (Hicks 1935, p. 456)

Hicks characterised both theories as “stationary”, and “quite satisfactory under that hypothesis, but incapable of extension to meet other hypotheses, and consequently incapable of application” (Hicks 1935, p. 456), because both theories assumed equalities that applied in a stationary state but could not be assumed in a changing one. Hicks warned that assuming such equalities where they did not exist was dangerous:

To found a theory upon an assumed equality, which is not a real equality, is a most dangerous thing to do; for the more complex the theory becomes, the more specialised it becomes. The blinkers grow, until they shut out nearly all the landscape. One distinction blurred over breeds another, until we have in the end only a special case of a special case of a special case. (Hicks 1935, p. 457)

Hicks therefore attempted to abandon the assumption of stationarity and develop a dynamic model. After advocating period analysis over the use of continuous time, Hicks set out his simplifying assumptions, which commenced with:

(1) We shall assume a community which is wholly engaged in the production of a single homogeneous good, which we shall call Bread.

(2) Bread is made by the co-operation of labour (assumed homogeneous) with capital goods (not homogeneous) which we shall call Equipment. Equipment may include land, buildings, machinery, raw materials, and half-finished goods. (Hicks 1935, p. 458)

These assumptions, in the context of a dynamic theory, require a model in which both bread and Equipment are produced—and in which raw materials, including energy, are exploited, as we model here. Had Hicks actually built this model, it would have been a true tour de force . Unfortunately, he did not. Instead, at a later stage in the paper, he reduced Equipment to dated bread:

A production plan can be regarded, on the basis of our simplifying assumptions, as a series of outputs of bread in successive weeks, together with the series of inputs of labour necessary to obtain those outputs. For the entrepreneur has actually to determine, not only how much labour he will employ in the first week, but how he will employ that labour, whether in the production of bread for the next market day, or in the production of bread for the more distant future

market"; for if the market for labour is in equilibrium, and if the market for bread is in equilibrium, the market for loans must be in equilibrium too .” (Hicks 1935, pp. 465-67. Emphasis added)

(activity which, a week after, will only have resulted in the production of equipment). (Hicks 1935, p. 460)

Hicks’s conceptual apparatus thus reduced to a model in which bread is produced using bread and labour alone, and in which bread functions as a consumer good if used this week, and a capital good if not used this week.[89]

When we first attempted to build a model which did achieve what Hicks set out to do, we felt genuine sympathy for his plight, since our attempt to build a model with the same conceptual foundation—an economy producing a single commodity, which functions as both a consumer and an investment good (which is a common feature of the vast majority of economic models, both Neoclassical and Post Keynesian)—led to a similar intellectual impasse. It is very easy to imagine a world in which consumers consume bread, but very difficult to imagine a world in which bread functions as machinery. In the end, Hicks’s sketch of a model described a passingly realistic scenario of consumption, but a trivial and unrealistic scenario for investment.

Our solution was to reverse this dilemma, and to consider a world with a far-fetched model of consumption, but a passingly realistic scenario for investment. What commodity can take the place “Applying these notions to the IS-LM construction, it is only the point of intersection of the curves which makes any claim to representing what actually happened (in our "1975"). Other points on either of the curves—say, the IS curve—surely do not represent, make no claim to represent, what actually happened. They are theoretical constructions, which are supposed to indicate what would have happened if the rate of interest had been different. It does not seem farfetched to suppose that these positions are equilibrium positions, representing the equilibrium which corresponds to a different rate of interest. If we cannot take them to be equilibrium positions, we cannot say much about them. But, as the diagram is drawn, the IS curve passes through the point of intersection; so the point of intersection appears to be a point on the curve; thus it also is an equilibrium position. That, surely, is quite hard to take. We know that in 1975 the system was not in equilibrium. There were plans which failed to be carried through as intended; there were surprises . We have to suppose that, for the purpose of the analysis on which we are engaged, these things do not matter. It is sufficient to treat the economy, as it actually was in the year in question, as if it were in equilibrium. Or, what is perhaps equivalent, it is permissible to regard the departures from equilibrium, which we admit to have existed, as being random. There are plenty of instances in applied economics, not only in the application of IS-LM analysis, where we are accustomed to permitting ourselves this way out. But it is dangerous. Though there may well have been some periods of history, some "years," for which it is quite acceptable, it is just at the turning points, at the most interesting "years," where it is hardest to accept it.” (Hicks 1981, pp. 149-50)

of bread, and enable a reasonably realistic model of production—including the use of raw materials and energy, and the production of machinery using that commodity as an input—at the probable expense of a rather unrealistic consumption good?

Fiction provided an answer with the cult animated movie The Iron Giant.[90] The deuteragonist of that movie was made of iron—see Figure 217. We therefore imagined a “Planet of the Iron Giants”, in which Iron Giants were the consumers and workers (and capitalists), iron was used to make the capital goods (blast furnace/rolling mill, iron ore and coal mining machines), energy was essential to all three processes, iron was consumed by the workers as their real wage, and physical waste (slag) was necessarily generated by production, as well as waste energy as in our previous model.

Figure 220: The "Iron Giant"

10.5 Derivation: constant technology and population

This was more easily said than done. To produce a model in which one commodity—iron—was the final output, we needed to model three sectors: the energy-mining sector (most easily thought of as coal mining, since coal—as coke—is also an input to iron manufacturing, and not solely an energy source); the iron-ore-mining sector; and a factory sector which took inputs of coal and iron ore to produce iron and slag. Each sector needed labour, and specialized capital—made of iron. Our mental framework was that everything was made of sheet iron, which could be shaped in the factory sector into shapes specific for each sector, and also as consumption for our Iron Giant workforce.

We needed three production relations, each with a different type of output, but each of which required energy (and capital and labour) as inputs. The outputs were respectively energy (best thought of as coal), iron ore, and iron plus slag.

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In keeping with Keen Ayres & Standish 2019, we treat machinery (“Capital”) as the means to channel energy to perform useful work. The output of an industry per year is the product of the number of machines K, times the energy per machine per year E, times the efficiency of conversion of energy into useful work ε , times the yield of product per unit of energy input y —this is the key extension over the previous model, where all internal processes in the model were in terms of energy only.

With three sectors (E for mining energy, M for mining iron ore, and F for factory), we have five equations: one for the output of each sector in units of energy per year (Joules/Year), units of iron ore per year (Kilograms/Year), units of physical output consisting of both iron and slag (Kilograms/Year), iron itself Y (our single commodity GDP), and slag YW (physical waste):

YW = ( 1 − µ) ⋅ F Mass/Year (Slag) Output that is used for investment adds to the stock of machines , which is denominated in terms of mass: kilograms of iron. This gives us a novel solution to the measurement of capital Y K problem: rather than ignoring the issue entirely as in standard Neoclassical models—despite Samuelson’s concession of defeat in the Cambridge Controversies (Sraffa 1960; Samuelson 1966; Pasinetti et al. 2003; Harcourt 1972)—or measuring capital in terms of dated labour as did Sraffa (Sraffa 1960), we measure capital in terms of kilograms of iron .[91] Using to signify the number of machines, and to signify the weight of each machine, we have: K k K = K ⋅ k + K ⋅ k + K ⋅ k

Employment is proportional to the number of machines in each sector. There is a workers per machine ratio such that employment in each industry equals this ratio times K: L λ LE = λ E ⋅ K E

This in turn enabled us to use the same wage change relation as in the previous models, based on the aggregate level of employment:

Three output equations are now needed, in contrast to earlier models with just one. A full, multicommodity-model would require price relations for each of the energy, iron mining and fabrication sectors, as well as stocks of unsold units of output. To generate a less complex single commodity model, we instead assumed proportionality between each sector, with excess capacity in energy and iron ore mining so that their yields adjust to meet the energy needs of the entire economy.[93] This means that the output of the energy sector equals to energy input needs of all three sectors: itself, mining, and fabrication:

This requires that the yield of the energy sector adjusts to the needs of all three sectors:

Solving for yields: E E

92 When the derivation of this model succeeded, we added a growing population and a falling labour to capital ratio in the final version, which is detailed in the next section. 93 This assumption is not a bad approximation to reality during a pre-ecological crisis period. It will be relaxed N in later extensions to allow analysis of falling EROEI or fossil to renewable energy switching.

The same assumption for mining, that the yield adjusts to fit the needs of the fabrication sector for material (iron ore) inputs, enables us to link the total output of the two sectors. Since the factory sector converts iron ore to rolled iron sheeting plus slag, then by the conservation of matter, the gross output of the factory in kilograms of iron plus slag equals the input in kilograms of iron ore. Therefore: F M

From this we can derive the yield (in kilograms per joule) of the factory sector:

Output from the factory sector can now be defined:

k M Call the first term—for the ratio between aggregate capital K and factory output (where this is the sum—in kilograms—of both iron and slag ) : F Y φ K = Y κ W M ⋅ φ E K Mk ⋅ε M ⋅ yM (159) M Then output and waste are: Y YW Y = µ⋅φ K ⋅ K

With output, labour and wages defined, it is now possible to derive the model in terms of the wages share of GDP and the employment rate. The rate of change of the wages share of GDP is a linear transformation of the rate of change of wages in this model without technical change or growth in population:

Therefore, the rate of change of wages share is a linear transformation of the wage change function:

The employment rate is a linear transformation of capital stock:

Hence

The final step in this process was to introduce a growing population and changing technology, manifest as a falling ratio of workers to machines. This in turn provides the scaffolding on which to add the accumulation of waste in the biosphere.

10.6 Growth and pollution

We replace a constant population with a growing one, and a constant labour to capital ratio with a falling one:

dt K κ K , κ A variable thus replaces 0 in (163) while becomes a variable in (161). The state space equation for thus becomes: K N N λ d 1 d ω ⋅

That for becomes: λ

This is once more the classic Goodwin model:

The strengths of this model over the previous versions are:

• The introduction of the concept of an “energy return on energy invested” in the yield from the energy mining sector, , since the input and the output are both energy. This is a critical concept in biophysical economics (Hall 2011), but, to our knowledge, has not E
• E
• previously been incorporated in a macroeconomic model.
• In this initial model, this is a constant derived from the requirement of the other two sectors. Our ambition is to make this an empirically derived quantity in future extensions, and to consider the extent to which this determines and constrains economic performance.
• • It is now possible to link this model directly to the generation of waste matter using equation (160): W
• Y

Figure 218 shows a simulation of this model in Minsky, including both output and waste derived from Equation (170).

Figure 221

This model thus achieves what Hicks attempted in "Wages and Interest: The Dynamic Problem" (Hicks 1935) in the context of a single commodity model of production. Future extensions will address the unrealism of this foundational model by introducing multiple commodities, and multiple forms of waste as well.

Footnotes

2. The fact that no quantitative change occurs by introducing energy into the Leontief production function, whereas a significant change occurs when doing the same with the Cobb-Douglas production function, indicates that the Leontief form was effectively correct, though based on a statistical regularity (the relatively constant capital to output ratio) rather than on energy; and

77 See (Blatt 1983) for an excellent explanation of this.

78 Though the word appeared only once in the text of these three papers: “The proposition that limited natural resources provide a limit to growth and to the sustainable size of population is an old one. The natural resource that was the centre of the discussion in Malthus' day was land; more recently, some concern has been expressed over the limitations imposed by the supplies of oil, or more generally, energy sources, of phosphorus, and of other materials required for production.” (Stiglitz 1974a, p. 123), and unlike Limits to Growth , no attempt was made to quantify either resources in general or energy in particular.

79 Given the ignorant and humourless state of economics in general, this isn’t a high bar: it’s more of a hurdle for sausage dogs. But Anwar cleared that bar by a large margin. Read the paper!

80 The Goodwin model’s empirically exaggerated variation in wages share is dramatically reduced when nonlinear behavioural functions, and monetary and price dynamics, are included in the model.

81 If the latter explanation for the rising ratio is more valid, then we should expect to see this ratio fall in the future if the dominance of the finance sector ever comes to an end.

84 This did exist, to some degree, in the “Computable General Equilibrium” models, but in mainstream economics these have largely been supplanted by Ramsey growth models, most of which use a Cobb-Douglas “production function”.

85 A worthwhile and highly publishable task for a motivated reader is to generalize this and make capacity utilization an endogenous variable of the model. This will create an (at least) 3-dimensional model, whose behaviour will be far more complex than that shown here.

88 “I must begin with the old story. "Mr. Keynes and the Classics" was actually the fourth of the relevant papers which I wrote during those years. The third was the review of The General Theory that I wrote for the Economic Journal, a first impression which had to be written under pressure of time, almost at once on first reading of the book. But there were two others that I had written before I saw The General Theory . One is well known, my "Suggestion for Simplifying the Theory of Money" (1935a), which was written before the end of 1934. The other, much less well known, is even more relevant. "Wages and Interest: the Dynamic Problem"' was a first sketch of what was to become the "dynamic" model of Value and Capital (1939). It is important here, because it shows (I think quite conclusively) that that model [IS-LM] was already in my mind before I wrote even the first of my papers on Keynes .” (Hicks 1981, p. 140. Emphasis added).

89 Hicks’s time period in "Wages and Interest: The Dynamic Problem" was a week, something which he later admitted made the Walrasian assumptions he made in 1935 inappropriate for the macroeconomic analysis of Keynes’s 1936 General Theory , which in 1937 he purported to capture with the IS-LM model. While it was appropriate in a week to consider expectations to be constant, it was not appropriate to consider the same when the time period is a year, because it implies constancy of expectations, which means the absence of surprises:

90 https://www.imdb.com/title/tt0129167/

91 Conceptually, the machines are rolled iron sheets molded into different shapes.